**9.1** The number of claims, *N*, has a binomial distribution with *m* = number of policies and *q* = 0.1. The claim amount variables, *X*_{1}, *X*_{2},…, all are discrete with Pr(*X*_{j} = 5,000) = 1 for all *j*.

**9.2** (a) An individual model is best because each insured has a unique distribution. (b) A collective model is best because each malpractice claim has the same distribution and there are a random number of such claims. (c) Each family can be modeled with a collective model using a compound frequency distribution. There is a distribution for the number of family members and then each family member has a random number of claims.

**9.3**

**9.4** The Poisson and all compound distributions with a Poisson primary distribution have a pgf of the form *P*(*z*) = exp{*λ*[*P*_{2}(*z*) − 1]} = [*Q*(*z*)]^{λ}, where *Q*(*z*) = exp[*P*_{2}(*z*) − 1].

The negative binomial and geometric distributions and all compound distributions with a negative binomial or geometric primary distribution have *P*(*z*) = {1 − *β*[*P*_{2}(*z*) − 1]}^{−r} = [*Q*(*z*)]^{r}, where *Q*(*z*) = {1 − *β*[*P*_{2}(*z*) − 1]}^{−1}.

The same is true for the binomial distribution and binomial-X compound distributions with *α* = *m* and *Q*(*z*) = 1 + *q*[*P*_{2}(*z*) − 1].

The zero-truncated and zero-modified distributions cannot be written in this form.

**9.5** To simplify writing the expressions, let

and similarly for *S*. For the first moment, , and ...

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